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A set-theoretic inclusion  $  V \subset  W $
of a linear [[Normed space|normed space]]  $  V $
into a linear normed space  $  W $, 
for which the following inequality is valid for any  $  x \in V $:

$$ 
\| x \| _ {W}  \leq   C  \| x \| _ {V} ,
$$

where  $  C $
is a constant which does not depend on  $  x \in V $. 
Here,  $  \| x \| _ {W} $
is the norm (semi-norm) of the element  $  x $
in  $  W $, 
while  $  \| x \| _ {V} $
is the norm (semi-norm) of the element  $  x $
in  $  V $.

The identity operator from  $  V $
into  $  W $, 
which assigns to an element  $  x \in V $
the same element seen as an element of  $  W $, 
is said to be the imbedding operator of  $  V $
into  $  W $. 
The imbedding operator is always bounded. If the imbedding operator is a [[Completely-continuous operator|completely-continuous operator]], the imbedding of function spaces is said to be compact. Facts on imbedding of function spaces are established by so-called [[Imbedding theorems|imbedding theorems]].

Example. Let  $  E $
be a Lebesgue-measurable set in the  $  n $-
dimensional Euclidean space with finite measure  $   \mathop{\rm mes}  E $
and let  $  L _ {p} ( E) $, 
$  1 \leq  p \leq  \infty $, 
be the Lebesgue space of measurable functions which are  $  p $-
th power summable over  $  E $
with norm

$$ 
\| x \| _ {p}  = \ 
\left [ \int\limits _ { E }
| x ( t) |  ^ {p}
 dt \right ]  ^ {1/p} .
$$

Then, if  $  p \geq  q $, 
one has the imbedding  $  L _ {p} ( E) \rightarrow L _ {q} ( E) $, 
and

$$ 
\| x \| _ {q}  \leq  \ 
(  \mathop{\rm mes}  E) ^ {1/q- 1/p }
\| x \| _ {p} .
$$

====Comments====
For references cf. [[Imbedding theorems|Imbedding theorems]].